A284148 Lexicographically earliest sequence of nonnegative integers such that a(1)=3 and the sequence is p-periodic mod p for any p > 0.

3, 0, 1, 0, 3, 28, 45, 276, 595, 1128, 1953, 3160, 4851, 264540, 190333, 254268, 18915, 3366496, 32385, 125391168, 199588483, 64620, 174673821, 5039370820, 1784859363, 16908230328, 165025, 34420237176, 58409997075, 1367074573228, 2294838551853, 15289788305820

Offset: 1

Author

Rémy Sigrist, Mar 21 2017

Keywords

• nonn

Comments

The initial term a(1)=3 seems to be the least one that leads to a sequence that does not have a polynomial closed form.

The first cycles mod p of this sequence are:

p Cycle of a mod p

- ----------------

1 0

2 1, 0

3 0, 0, 1

4 3, 0, 1, 0

5 3, 0, 1, 0, 3

6 3, 0, 1, 0, 3, 4

7 3, 0, 1, 0, 3, 0, 3

8 3, 0, 1, 0, 3, 4, 5, 4

9 3, 0, 1, 0, 3, 1, 0, 6, 1

For k>=0, let c_k denote the variant with initial term k.

Naturaly, we have a=c_3.

For some values of k, c_k has a polynomial closed form.

The first such values to be known are:

- k=0: c_0(n) = 0 = A000004(n),

- k=1: c_1(n) = (n-2)^2 = A000290(n-2),

- k=2: c_2(n) = (n-2)*(n-3) = A002378(n-3),

- k=19: c_19(n) = (n-2)*(n^3 - 14*n^2 + 63*n - 88)/2,

- k=20: c_20(n) = (n-2)*(n-3)*(n-5)*(n-6)/2,

- k=22: c_22(n) = (n-2)*(n-3)*(n^2 - 11*n + 32)/2,

- k=40: c_40(n) = (n-2)*(n-3)*(n-5)*(n-6),

- k=172: c_172(n) = (n-2)*(n-3)*(n-5)*(n^3 - 23*n^2 + 172*n - 408)/12.

We notice that c_40 = 2*c_20.

As for A281409, this sequence is the first of a family (of sequences parametrized by their initial term) showing some kind of irregularity.

For k>=0 and n>0, let d_n(k)=c_k(n):

- In particular: d_1(k)=k, and a(n)=d_n(3),

- For any n>1, d_n is periodic.

The cycles for the first d_n (with n>1) are:

n Cycle of d_n

- ------------

2 0

3 0, 1

4 0, 4, 2

5 0, 9, 6, 3

6 0, 16, 12, 28, 24, 40, 36, 52, 48, 4

7 0, 25, 20, 45, 40, 5

Examples

			By definition, a(1)=3.
a(2) must equal 3 mod 1; a(2)=0 is suitable.
a(3) must equal 3 mod 2 and 0 mod 1; a(3)=1 is suitable.
a(4) must equal 3 mod 3 and 0 mod 2 and 1 mod 1; a(4)=0 is suitable.
a(5) must equal 3 mod 4 and 0 mod 3 and 1 mod 2 and 0 mod 1; a(5)=3 is suitable.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..2000
• Rémy Sigrist, PARI program for A284148

Crossrefs

Cf. A000004, A000290, A002378, A281409.